Bolzano’s theorem and the Archimedian property of R.

Use Bolzano’s theorem and the supremum to prove the Archimedian property of R.
I know bolzano's theorem shows that supA exists and the Arc.property of R shows that n>x.. but how do I use those two methods to prove one another?

The archimedian property is: For any x, there exists a natural number n with n>x.

You can prove that R has the archimedian property this way:
Let x>0 and
A={n : n is a natural number and n<=x}
Then A is bounded above by x and 0\in A,
so sup A=s exists.
Since s is the least upper bound of A, s-1 is not an upper bound of A anymore.
So there exists n \in A with s-1<n
=>
s<n+1
Since s is an upper bound of A
=> n+1 \notin A
=>
x<n+1